For any open set \(\Omega \subset \mathbb {R}^2,\)  let \(\mathcal {D}(\Omega )\)  be the space of compactly supported smooth functions on \(\Omega \) . The space of distributions \(\mathcal {D}^{\prime }(\Omega )\)  is defined as the dual space of \(\mathcal {D}(\Omega )\)  [1]}. Similarly, \(\mathcal {D}^{\prime }(\Omega ,\mathbb {R}^2)\)  and \(\mathcal {D}^{\prime }(\Omega ,\operatorname{Lin})\)  are the spaces of vector and tensor valued distributions, respectively, which are dual to the spaces of compactly supported smooth functions \(\mathcal {D}(\Omega ,\mathbb {R}^2)\)  and \(\mathcal {D}(\Omega ,\operatorname{Lin})\) , respectively. Any locally integrable function \(f\)  can be associated with the distribution \(T_f \in \mathcal {D}^{\prime }(\Omega )\)  given by \(T_f (\psi )=\int _\Omega f \psi \operatorname{da}\)  for all \(\psi \in \mathcal {D}(\Omega ).\)  We say that a distribution \(T\in \mathcal {D}^{\prime }(\Omega )\)  is continuous (or smooth) if there exists a continuous (or smooth) function \(f\)  such that \(T (\psi )=\int _\Omega f \psi \operatorname{da}\)  for all \(\psi \in \mathcal {D}(\Omega )\) .
